Problem 93 - Calculus for Biology and Medicine

Question

Drug Absorption After a patient takes the painkiller acetaminophen (often sold under the brand name Tylenol), the concentration of drug in their blood increases at first, as the painkiller is absorbed into the blood, and then starts to decrease as the drug is metabolized or removed by the liver. In one study, the concentration of drug \((c\), measured in \(\mu \mathrm{g} / \mathrm{ml}\) ) was measured in a patient as a function of time \((t\), measured in hours since the drug was administered). The data in this example is taken from Rowling et al. (1977). \begin{tabular}{lc} \hline \(\boldsymbol{t}\) & \multicolumn{1}{c} {\(\boldsymbol{c}\)} \\ \hline 1 & \(10.61\) \\ \(1.5\) & \(8.73\) \\ 2 & \(7.63\) \\ 3 & \(5.55\) \\ 4 & \(3.97\) \\ 5 & \(3.01\) \\ 6 & \(2.39\) \\ \hline \end{tabular} (a) You want to determine from the data whether the relationship between concentration and time follows a power law $$ c=a t^{b} $$ for some set of constants \(a\) and \(b\), or whether it instead follows an exponential law $$ c=k d^{t} $$ for some constants \(k\) and \(d\). Explain how you could plot the data with transformed horizontal and vertical axes to determine which mathematical model is correct. (b) By plotting \(\log c\) against \(\log t\) in one graph, and \(\log c\) against \(t\) in another, explain why the data supports the second model (exponential decay) better than it supports the first model. (c) From your plot of \(\log c\) against \(t\), estimate the parameter \(d\).

Step-by-Step Solution

Answer

Use transformed plots to test each model: the \(\log c\) vs. \(t\) plot supporting a straight line indicates exponential decay. Estimate \(d\) from the slope of this plot.

Step 1: Understanding the Models

A power law model is represented by the equation \(c = a t^b\), which can be linearized by taking the logarithm of both sides, resulting in \(\log c = \log a + b \log t\). Here, plotting \(\log c\) vs. \(\log t\) should yield a straight line if this model is correct. Conversely, an exponential model is represented by the equation \(c = k d^t\), which upon taking the logarithm of both sides, becomes \(\log c = \log k + t \log d\). In this case, plotting \(\log c\) vs. \(t\) should produce a straight line.

Step 2: Data Transformation for Plotting

Transform the data for both potential models. Compute \(\log c\) for each concentration value and \(\log t\) for each corresponding time value. For example, if \(c = 10.61\) and \(t = 1\), then \(\log c = \log 10.61\) and \(\log t = \log 1 = 0\). Repeat for all data points.

Step 3: Plotting for Power Law Model

Plot the transformed data with \(\log t\) on the x-axis and \(\log c\) on the y-axis. If the data forms a straight line, it suggests the relationship is a power law.

Step 4: Plotting for Exponential Model

Plot the transformed data with \(t\) on the x-axis and \(\log c\) on the y-axis. This plot helps to see if an exponential model fits the data, indicating a straight line under this transformation suggests exponential decay.

Step 5: Analyzing Plots

Compare the linearity of the two plots. A better-fitting line in the \(\log c\) vs. \(t\) plot (compared to the \(\log c\) vs. \(\log t\) plot) supports the exponential model over the power model.

Step 6: Estimating Exponential Parameter d

Using the \(\log c\) vs. \(t\) plot, determine the slope, which corresponds to \(\log d\). If \(m\) is the slope from this plot, then \(d = 10^m\). Measure the slope from the plotted line to estimate \(d\).

Key Concepts

Power Law Model

In drug absorption modeling, a power law model is represented by the equation \(c = a t^b\). This model suggests that the relationship between drug concentration \(c\) and time \(t\) follows a power function, characterized by constants \(a\) and \(b\). To verify if the data follows a power law, we can transform the original relationship into a linear one by taking the logarithm of both sides. This gives us:\[\log c = \log a + b \log t\]By plotting \(\log c\) against \(\log t\), we should observe a straight line if the power law model is a good fit. The slope of this line is \(b\), and the intercept is \(\log a\).

• This approach provides a clear way to check for power law behavior.
• If the plot does not produce a linear pattern, the power law model may not be suitable.

Exponential Decay

Exponential decay refers to a process where the quantity decreases at a rate proportional to its current value. In the context of drug absorption, this is modeled by \(c = k d^t\), where \(k\) is the initial concentration, and \(d\) is the base of the exponential decay. To linearize this relationship, we take the logarithm of both sides to obtain:\[\log c = \log k + t \log d\]Plotting \(\log c\) against \(t\) should yield a straight line if the data follows an exponential decay model. The slope of this straight line is \(\log d\), and the intercept is \(\log k\).

• This form provides an easy way to identify exponential decay behavior.
• The linearity of the \(\log c\) vs. \(t\) plot confirms the suitability of the exponential model.

Data Transformation

Data transformation is a crucial step in analyzing relationships between variables. For drug absorption data, transforming the values of concentration and time makes it easier to evaluate the two proposed models. By converting concentration \(c\) values to \(\log c\) and time \(t\) values to \(\log t\), we can better visualize and assess linearity in plots.

• Transformation streamlines complex mathematical models to linear forms.
• These linearized transformations are simpler to interpret, helping in model assessment.

For each data point in the study, compute \(\log c\) and \(\log t\) to see which model best describes the data when plotted.

Plot Analysis

Plot analysis is essential in determining the correct model for data. By creating and analyzing plots, we can visually assess how well a model fits the data. For this exercise, two specific plots are useful:

• Plot \(\log c\) vs. \(\log t\) for the power law model.
• Plot \(\log c\) vs. \(t\) for the exponential decay model.

If the data points closely align along a straight line in the \(\log c\) vs. \(t\) plot, it suggests that the exponential decay model is a better fit. This is because the linear nature in this transformation indicates decay proportional to the current concentration, a hallmark of exponential behavior.Choosing the Right Model:

• The slope of the \(\log c\) vs. \(t\) plot provides an estimate for \(\log d\).
• Using this slope, calculate \(d = 10^m\) where \(m\) is the slope, confirming the model's parameters.